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On the “new” 16 + 16 version of N = 1 supergravity
Volume 172, number 3,4
PHYSICS LETTERS B
22 May 1986
ON THE "NEW" 16 + 16 V E R S I O N OF N - - 1 S U P E R G R A V I T Y
M. H A Y A S H I Department of Physics, Kyoto University, Kyoto 606, Japan
and S. U E H A R A 1 Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima 725, Japan Received 27 November 1985; revised manuscript received 16 January 1986
It is shown explicitly that the recently proposed 16 + 16 supergravity is derived using both a chiral and a real linear multiplet as compensator from N = 1 conformal supergravity. Thus this version is reducible.
It is now well known that there exist three different irreducible off-shell formulations o f N = 1 Poincar6 supergravity;old minimal (12 + 12) [1], a new minimal (12 + 12) [2] and non-minimal (20 + 20) [3]. (Here (n + n) denotes the numbers of bosonic and fermionic degrees of freedom.) These three versions are systematically derived using compensating multiplets one by one from the unique N = 1 conformal supergravity [4]. Recently a "new" type of formulation has been proposed which has 16 + 16 degrees of freedom and claimed to be irreducible [5]. This version contains, besides a graviton and a gravitino ~ u ' a real vector Gu' a complex scalar S, a real scalar 4, a real second-rank antisymmetric tensor Nuv (or divergenceless vector Wu) and a Majorana spinor field T. Gu and g are just auxiliary while ¢,Nta v and T propagate in the case o f n < -~- or 0 < n, where n is a parameter in this version (hereafter we regard n < - ~ or 0 < n without mention). If we take the limit n -+ 0 properly, this version becomes the new minimal version, while in the case of n ~ -½ it becomes the old minimal version. So this version seems a hybrid of the old minimal and the new minimal ones. In this letter we derive this version explicitly from the N = 1 conformal supergravity and show that it is, in fact, reducible. The compensating multiplet to derive the 16 + 16 version is reducible and consists of a chiral multiplet ~2(w-n=l) and a real linear multiplet t : I~(w=n=l) = ( A , B , x , F , G ) , L =(C,Z,Bm),
(1)
with D C B m = 0 .
(2)
Here A, B, F, G and C are real scalar fields, X and Z are Majorana spinors and B m is a real conserved vector field which can be written by a second-rank antisymmetric tensor field auv [see ref. [6] for notation and conversions]. The D, A, S and K m gauge transformations are fixed by the following conditions: A - 6 n c 3n+1 = 1 ,
D gauge,
B = 0,
A gauge,
(3)
1 This work was supported in part by the Grant-in-Aid for Scientific Research, Ministry of Education, Science and Culture, Japan (#60740138).
348
Volume 172, number 3,4
PHYSICS LETTERSB
(3n + 1)Ai3,5 Z - 6nCx = 0 ,
S gauge,
bet = O,
K m gauge,
22 May 1986
(3 cont'd)
where bu is the D gauge field. This "mixing-up" gauge fixing condition leads to the seeming "irreducibility" of the reduced Poincar~ supergravity. This gauge can be transformed by a field-dependent gauge transformation into such a gauge where reducibility is manifest [7,8]. Since both chiral and real linear multiplets have 4 + 4 degrees of freedom, the reduced Poincard supergravity has 16 + 16 degrees of freedom. The supersymmetry transformation in Poincar~ supergravity, which preserves the gauge-f'txing condition (3), is given by = ~Q(~) + 6S(,/) + ~K(41- g~m --~ V?~m) + ~A {-- [( 3n + 1 ) / 6 n ] C - l ~ z } ,
~(e)
r/--- - [(3n + 1)(3n - 1)/24n]C-275Z(g75 Z) + [(3n + 1)2/24n]C-2Z(gZ) 3
-- ~ i n {'4rn -- [(3n + 1)/3n ] C - 1Bm } 75 7m ~ + 23-n C - ( 3n + 1)/6 n (F + i75 G) t ,
Am =Am + [(3n + 1)/6n]C-l~m z ,
(4)
where q~t is the dependent S gauge field (see ref. [6]). Each field in ref. [5] is identified as follows: C=exp(---4mk),
Z = - - 4 n e x p ( - 4 n ~ ) i 7 5 T,
Bm=-Wm-12in2exp(--4ntk)
TLTmTR,
5r=gl ( F + i G ) =1~_e x p [ - } ( 3 n + 1 ) f f ] [ S - ~-(3n + 1) TRTR] , -- = ½ ( F - i G ) = g l
e x p [ - } ( 3 n + 1 ) ¢ ] [ S - ~ 4 ( 3 n + 1)TLTL] ,
"4m = (2/3n) Gm _ [(3n + 1)/3n] exp(4n¢) I¢m - 4~ i(3n + 1)(n + 1) TL3'mTR ~v = - ~ u '
T R = ~ '(I + 75)T,
T L = } ( 1 - 75)T.
(5)
From eqs. (4) and (5) the transformation laws for the independent fields can be obtained
1
8~(e)TR=~-eR[~+( n_I)~RTR]_~TmeL[D + g1
coy,
m
$ + ( i / 2 n ) ~ m _ ( i / 4 n ) exp(4nt~)Wm]
(5n + 1) T R ( T L e L ) ,
6 ~ ( e ) Wm = e x p ( - 4 n ~ ) ( 6in(n 2 + 4n + 1)(TL7 m TR)(Te ) + 4in2DCOV.n~(T'Y5amne ) - 6in2DC°V.~,(~75 e ) + 4in~3, 50mn Dc°v'n T+ } igOmnTl(7 st} e°v-"l + } ~ c o v . , l ) + 2[G n - ½ exp(4nff) W n] TOmn e 3
+ ~- n exp(4nO) Wm (Te)} , 8(~(e)g = (3n + 1)S(T75¢) + [4(3n + 1)(n 2 + 4n. + 1)TLeL - ~-(3n + 1)(3n + 5)TReR] + 2(3n + 1)~c°v" T R -- gL
omn~bc°v" "r m n , L
+4n(3n + 1) {Dm coy. ff _ (i/4n2) Gm +i[(n + 1)/8n] exp(4nt~)Wm}TR'yme L ,
(6) 349
Volume 172, number 3,4
PHYSICS LETTERS B
22 May 1986
6 ~ ( ¢ ) 6 m = in(3n + 1) g757mD c°v" T + 8in2(3n + 1)D c°v.n ~ T 7 5 a m n - in(3n + 1)(2n - 1) T757m ( D c ° v ' ~ ) t + 2(3n + 1)Gn(T°mn E) 1 (3n + l)(n + 2) exp(4nff) Wn(TOmn ¢) + ~ n(3n + 1) exp(4mk) Wm (T¢) + ~-in(3n + 1)[S(~LTmtR) - - S ( ~ R T m t L ) ] + 2in(3n + 1)(2n 2 + 10n + 3 ) ( T L T m T R ) ( T ~ ) --
1 i n g ~1 5 1 ~.n,7,cov. I i(2n + 1) g 7 n ~ mn c°v" , 'emn -- 4
2
5~(¢) ~ . R = -- [ D ~ R
-- 2 n 2 7 . T L ( T R £ R ) + n 2 t R TLTt~TR] + 1 n T . ¢ L ~
+ ~ i~-R ( ( l / n ) G . - [(3n + 1)/2hi exp(4n~) Wu ) + (2n + 1)2gR(TLT.TR) + ~ i7.~E R + 2n(2n + 1) 7 . T L ( T R ¢ R) + ½ (3n + 1)[tR (T75 ~u) - ~ . R ( T 7 5 ¢ ) ] , 8PQ(g)e~ m = - ~ gTm ~t~'
(6 cont'd)
where * ~cov. - . v = DutO'~kv + ½(3n + 1) 75 ~ ( T 7 5 ~kv) + 0/2n) 75 ~ . [Gv - ½ (3n + 1) exp(4nff) Wu - 2in(3n + 1)(n + 1)(TL7 v TR) ]
-~i757ta mn
1_
~n(STta~vR +S)'u~vL) S, ov kl
z mnklW
"
,
--m
cov._
tO
=D m -8
2 n ( 3 n + 1)[7~TR(TL~vL)+71aTL(TR~t,R)] (-~m),
tO
Dm--em#
[0.
--(fl'~'v),
-
The lagrangian is given by £ = (1/2n)[(~ @~*)-3n X (V(L))3n+I]D ,
(7)
where V (L) means that L is embedded into a real vector multiplet; V(L) = [C,Z, 0, O, B m , - ~ c z , -l-ICe] .
(8)
Note that when n = - ~ , £ becomes ./~(n = - ~ ) = - ~
[Z @ ]~ *]D ,
(9)
which leads to the old minimal supergravity. On the other hand, if we take the limit n -+ 0, £ becomes £ = ( 1 / 2 n ) [ V ( L ) (1 + 3n ln[V (t.)/I~ ®~c*] + O(n2)}]D n ~
~ [V(L) × l n t V ( t ) / l ~ ® Z*]] D ,
(10)
which leads to the new minimal supergravity [c.f. [V(L)] D = 0]. The lagrangian (7) is represented by component fields as follows:
,1 Our spin connection tot~kl becomes -t%kl(e, 4) due to (3) and then it is slightly different from the one in ref. [5 ]. 350
Volume 172, number 3,4
PHYSICS LETTERS B
22 May 1986
e - l £ = - ~ cl~ - ~ e-l~.t~vP°4~T57v(D~t~ o + n2T5oph~a TT57~'T) - 4n(3n + 1) T7 m (D~nT + n275 Omn TT75 TnT) - 4n(3n + 1)(~ m 4)(~m 4)
+ nSS - (1/n)GmGm + [(3n + 1)/4n] exp(8n4) W2m - 2i(2n + 1)(3n + 1)(TR7 m TL) exp(4n4w m) + i(3n + 1)(TT5 ~m) exp(4n4w m) + 4n(3n + 1)(~nTm'TnT)Om 4 -
8n(3n + 1)(9n 2 + 5n + 1)(TRTR)(TLTL)
-- 2n(2n + 1)(3n + 1)[(t~R.rTL)(TRTR) + (t~L .TTR)(TLTL) ]
+ ~ n(3n + 1) [(t~rn R ~ ) ( T R TR) + (t~mL ~Ln)(TLTL)] -- 2n(3n + 1)(t~m R TR)(~[n TL) + 2n (7n + 2)[(~m
R
crmn TR)(t~n L TL) + (~mL °ran TL)(~n RTR)] •
(11)
Since the "new" 16 + 16 version is derived with a reducible compensating multiplet, this version is reducible. The seeming irreducibility comes from the "mixing-up" gauge condition (3) and the particular definition of fields (5). Finally we give explicitly the field-dependent gauge transformation which changes the "mixing-up" gauge conditions into the ordinary gauge conditions of the old minimal version [8]. Let us introduce C O and Z 0 fields,
C O = CA -2 ,
Z 0 =Z + 2CiT5 (A + i T 5 B ) - I x .
(12)
An advantage of using three variables is that C O is D, S, K-inert and Z 0 is S, K-inert. Then the desired gauge transformation is given by ,2 u = UKUSUD,
U D = exp [a D (/9)1 ,
with p = ~ (3n + 1) In C O ,
U S = exp [a s (~')] ,
with ~"= ~(3n + 1 ) c - l i T 5 Z 0 ,
U K = exp[6K(~m)] , with 7/m = e m (3 ~uP + ~ ~ ) .
(13)
The gauge fixing conditions (3) become
A' = UAU -1 = exp(p)A = 1,
B' = UBU -1 = exp(p)B = 0 ,
r
X =UxU - l = e x p ( ~ p ) [ X + ( A + i 7 5 B ) ~ " ] = 0 ,
t
b~=Ub~U -l=bu+aup+
1
~-2e~m~m
=0.
(14)
And other independent fields are redefined as follows : C' = e x p ( ~ )
Z ' = ~4- e x p [ 1~ ( 3 n + 5 ) 4 ] i 7 5 T,
,
B m = exp[2(3n + 1)4] [-W m - ] i exp(--4n4)(TLTm TR)] , P
A~ = (2/3n)G~
__
5r' = ~ exp[~(3n + 1)4] [S - ~-(3n + 1)TRTR] ,
[(3n + 1)/3n] e x p ( a n 4 ) Wu - ~ i(3n + 1) TLT~TR
4'u=exp[-~(3n+l)4][-~u+](3n+l)T~T],
e, m ' = e x p [ - ] ( 3 n + l ) 4 ] e g m
(15)
,2 Hexewe use the usual symbols 6D,S,K(.) although the parameters are not infinitesimal. 351
Volume 172, number 3,4
PHYSICS LETTERS B
22 May 1986
The remaining task is to rewrite the Poincar~ supersymmetry transformation 6~ ( t ) on the basis of the primed fields ¢ ' = UCU - 1 . The result is
~PQ(f,) = ~ld(Co(3n+l)14 ~,) + 6~lld (~-(3n
+ 1)C - 1
gi75omnZo).
(16)
Here 6~ ld is the usual Poincar6 supersymmetry transformation on the primed fields in the old minimal case. So eq. (16) explicitly shows that the complicated transformation laws 6~ ( t ) o f eq. (6) is in fact reducible to that o f the old minimal case if we rewrite all the fields by the primed ones. And the lagrangian (11) rewritten in terms o f the primed fields (15) shows that the real linear multiplet [C', Z ' , B m ] couples to the old minimal supergravity We would like to thank T. Kugo for valuable discussions.
Note added. After completing the manuscript we were informed by T. Kugo about a preprint [9] in which the reducibility o f 16 + 16 supergravity is pointed out. After submitting this paper, we received a preprint [10] from one o f the authors. One o f their gauge fixing conditions (8b) is quite different but the difference does not affect the reduced Poincar~ supergravity transformation laws and the lagrangian. References [1] K.S. Stelle and P.C. West, Phys. Lett. B 74 (1978) 330; S. Ferrara and P. van Nieuwenhuizen, Phys. Lett. B 74 (1978) 333. [2] M.F. Sohnius and P.C. West, Phys. Lett. B 105 (1981) 353. [3] P. Breitenlohner, Phys. Lett. B 67 (1977) 49; Nucl. Phys. B 124 (1977) 500. [4] M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Phys. Rev. D17 (1978) 3179; M. Kaku and P.K. Townsend, Phys. Lett. B 76 (1978) 54; P.K. Townsend and P. van Nieuwenhuizen, Phys. Rev. D19 (1979) 3166, 3592. [5] G. Girardi, R. Grimm, M. MiiUer and J. Wess, Phys. Lett. B 147 (1984) 81. [6] See for example: T. Kugo and S. Uehara, Nucl. Phys. B226 (1983) 49; T. Kugo and S. Uehara, Prog. Theor. Phys. 73 (1985) 235. [7] B. de Wit, R. Philippe and A. Van Proeyen, Nucl. Phys. B 219 (1983) 143. [8] T. Kugo and S. Uehara, Nucl. Phys. B226 (1983) 93. [9] W. Siegel, Maryland preprint #86-48. [10] C.S. Aulakh, J.-P. Derendinger and S. Ouvry, Paris preprint.
352
PHYSICS LETTERS B
22 May 1986
ON THE "NEW" 16 + 16 V E R S I O N OF N - - 1 S U P E R G R A V I T Y
M. H A Y A S H I Department of Physics, Kyoto University, Kyoto 606, Japan
and S. U E H A R A 1 Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima 725, Japan Received 27 November 1985; revised manuscript received 16 January 1986
It is shown explicitly that the recently proposed 16 + 16 supergravity is derived using both a chiral and a real linear multiplet as compensator from N = 1 conformal supergravity. Thus this version is reducible.
It is now well known that there exist three different irreducible off-shell formulations o f N = 1 Poincar6 supergravity;old minimal (12 + 12) [1], a new minimal (12 + 12) [2] and non-minimal (20 + 20) [3]. (Here (n + n) denotes the numbers of bosonic and fermionic degrees of freedom.) These three versions are systematically derived using compensating multiplets one by one from the unique N = 1 conformal supergravity [4]. Recently a "new" type of formulation has been proposed which has 16 + 16 degrees of freedom and claimed to be irreducible [5]. This version contains, besides a graviton and a gravitino ~ u ' a real vector Gu' a complex scalar S, a real scalar 4, a real second-rank antisymmetric tensor Nuv (or divergenceless vector Wu) and a Majorana spinor field T. Gu and g are just auxiliary while ¢,Nta v and T propagate in the case o f n < -~- or 0 < n, where n is a parameter in this version (hereafter we regard n < - ~ or 0 < n without mention). If we take the limit n -+ 0 properly, this version becomes the new minimal version, while in the case of n ~ -½ it becomes the old minimal version. So this version seems a hybrid of the old minimal and the new minimal ones. In this letter we derive this version explicitly from the N = 1 conformal supergravity and show that it is, in fact, reducible. The compensating multiplet to derive the 16 + 16 version is reducible and consists of a chiral multiplet ~2(w-n=l) and a real linear multiplet t : I~(w=n=l) = ( A , B , x , F , G ) , L =(C,Z,Bm),
(1)
with D C B m = 0 .
(2)
Here A, B, F, G and C are real scalar fields, X and Z are Majorana spinors and B m is a real conserved vector field which can be written by a second-rank antisymmetric tensor field auv [see ref. [6] for notation and conversions]. The D, A, S and K m gauge transformations are fixed by the following conditions: A - 6 n c 3n+1 = 1 ,
D gauge,
B = 0,
A gauge,
(3)
1 This work was supported in part by the Grant-in-Aid for Scientific Research, Ministry of Education, Science and Culture, Japan (#60740138).
348
Volume 172, number 3,4
PHYSICS LETTERSB
(3n + 1)Ai3,5 Z - 6nCx = 0 ,
S gauge,
bet = O,
K m gauge,
22 May 1986
(3 cont'd)
where bu is the D gauge field. This "mixing-up" gauge fixing condition leads to the seeming "irreducibility" of the reduced Poincar~ supergravity. This gauge can be transformed by a field-dependent gauge transformation into such a gauge where reducibility is manifest [7,8]. Since both chiral and real linear multiplets have 4 + 4 degrees of freedom, the reduced Poincard supergravity has 16 + 16 degrees of freedom. The supersymmetry transformation in Poincar~ supergravity, which preserves the gauge-f'txing condition (3), is given by = ~Q(~) + 6S(,/) + ~K(41- g~m --~ V?~m) + ~A {-- [( 3n + 1 ) / 6 n ] C - l ~ z } ,
~(e)
r/--- - [(3n + 1)(3n - 1)/24n]C-275Z(g75 Z) + [(3n + 1)2/24n]C-2Z(gZ) 3
-- ~ i n {'4rn -- [(3n + 1)/3n ] C - 1Bm } 75 7m ~ + 23-n C - ( 3n + 1)/6 n (F + i75 G) t ,
Am =Am + [(3n + 1)/6n]C-l~m z ,
(4)
where q~t is the dependent S gauge field (see ref. [6]). Each field in ref. [5] is identified as follows: C=exp(---4mk),
Z = - - 4 n e x p ( - 4 n ~ ) i 7 5 T,
Bm=-Wm-12in2exp(--4ntk)
TLTmTR,
5r=gl ( F + i G ) =1~_e x p [ - } ( 3 n + 1 ) f f ] [ S - ~-(3n + 1) TRTR] , -- = ½ ( F - i G ) = g l
e x p [ - } ( 3 n + 1 ) ¢ ] [ S - ~ 4 ( 3 n + 1)TLTL] ,
"4m = (2/3n) Gm _ [(3n + 1)/3n] exp(4n¢) I¢m - 4~ i(3n + 1)(n + 1) TL3'mTR ~v = - ~ u '
T R = ~ '(I + 75)T,
T L = } ( 1 - 75)T.
(5)
From eqs. (4) and (5) the transformation laws for the independent fields can be obtained
1
8~(e)TR=~-eR[~+( n_I)~RTR]_~TmeL[D + g1
coy,
m
$ + ( i / 2 n ) ~ m _ ( i / 4 n ) exp(4nt~)Wm]
(5n + 1) T R ( T L e L ) ,
6 ~ ( e ) Wm = e x p ( - 4 n ~ ) ( 6in(n 2 + 4n + 1)(TL7 m TR)(Te ) + 4in2DCOV.n~(T'Y5amne ) - 6in2DC°V.~,(~75 e ) + 4in~3, 50mn Dc°v'n T+ } igOmnTl(7 st} e°v-"l + } ~ c o v . , l ) + 2[G n - ½ exp(4nff) W n] TOmn e 3
+ ~- n exp(4nO) Wm (Te)} , 8(~(e)g = (3n + 1)S(T75¢) + [4(3n + 1)(n 2 + 4n. + 1)TLeL - ~-(3n + 1)(3n + 5)TReR] + 2(3n + 1)~c°v" T R -- gL
omn~bc°v" "r m n , L
+4n(3n + 1) {Dm coy. ff _ (i/4n2) Gm +i[(n + 1)/8n] exp(4nt~)Wm}TR'yme L ,
(6) 349
Volume 172, number 3,4
PHYSICS LETTERS B
22 May 1986
6 ~ ( ¢ ) 6 m = in(3n + 1) g757mD c°v" T + 8in2(3n + 1)D c°v.n ~ T 7 5 a m n - in(3n + 1)(2n - 1) T757m ( D c ° v ' ~ ) t + 2(3n + 1)Gn(T°mn E) 1 (3n + l)(n + 2) exp(4nff) Wn(TOmn ¢) + ~ n(3n + 1) exp(4mk) Wm (T¢) + ~-in(3n + 1)[S(~LTmtR) - - S ( ~ R T m t L ) ] + 2in(3n + 1)(2n 2 + 10n + 3 ) ( T L T m T R ) ( T ~ ) --
1 i n g ~1 5 1 ~.n,7,cov. I i(2n + 1) g 7 n ~ mn c°v" , 'emn -- 4
2
5~(¢) ~ . R = -- [ D ~ R
-- 2 n 2 7 . T L ( T R £ R ) + n 2 t R TLTt~TR] + 1 n T . ¢ L ~
+ ~ i~-R ( ( l / n ) G . - [(3n + 1)/2hi exp(4n~) Wu ) + (2n + 1)2gR(TLT.TR) + ~ i7.~E R + 2n(2n + 1) 7 . T L ( T R ¢ R) + ½ (3n + 1)[tR (T75 ~u) - ~ . R ( T 7 5 ¢ ) ] , 8PQ(g)e~ m = - ~ gTm ~t~'
(6 cont'd)
where * ~cov. - . v = DutO'~kv + ½(3n + 1) 75 ~ ( T 7 5 ~kv) + 0/2n) 75 ~ . [Gv - ½ (3n + 1) exp(4nff) Wu - 2in(3n + 1)(n + 1)(TL7 v TR) ]
-~i757ta mn
1_
~n(STta~vR +S)'u~vL) S, ov kl
z mnklW
"
,
--m
cov._
tO
=D m -8
2 n ( 3 n + 1)[7~TR(TL~vL)+71aTL(TR~t,R)] (-~m),
tO
Dm--em#
[0.
--(fl'~'v),
-
The lagrangian is given by £ = (1/2n)[(~ @~*)-3n X (V(L))3n+I]D ,
(7)
where V (L) means that L is embedded into a real vector multiplet; V(L) = [C,Z, 0, O, B m , - ~ c z , -l-ICe] .
(8)
Note that when n = - ~ , £ becomes ./~(n = - ~ ) = - ~
[Z @ ]~ *]D ,
(9)
which leads to the old minimal supergravity. On the other hand, if we take the limit n -+ 0, £ becomes £ = ( 1 / 2 n ) [ V ( L ) (1 + 3n ln[V (t.)/I~ ®~c*] + O(n2)}]D n ~
~ [V(L) × l n t V ( t ) / l ~ ® Z*]] D ,
(10)
which leads to the new minimal supergravity [c.f. [V(L)] D = 0]. The lagrangian (7) is represented by component fields as follows:
,1 Our spin connection tot~kl becomes -t%kl(e, 4) due to (3) and then it is slightly different from the one in ref. [5 ]. 350
Volume 172, number 3,4
PHYSICS LETTERS B
22 May 1986
e - l £ = - ~ cl~ - ~ e-l~.t~vP°4~T57v(D~t~ o + n2T5oph~a TT57~'T) - 4n(3n + 1) T7 m (D~nT + n275 Omn TT75 TnT) - 4n(3n + 1)(~ m 4)(~m 4)
+ nSS - (1/n)GmGm + [(3n + 1)/4n] exp(8n4) W2m - 2i(2n + 1)(3n + 1)(TR7 m TL) exp(4n4w m) + i(3n + 1)(TT5 ~m) exp(4n4w m) + 4n(3n + 1)(~nTm'TnT)Om 4 -
8n(3n + 1)(9n 2 + 5n + 1)(TRTR)(TLTL)
-- 2n(2n + 1)(3n + 1)[(t~R.rTL)(TRTR) + (t~L .TTR)(TLTL) ]
+ ~ n(3n + 1) [(t~rn R ~ ) ( T R TR) + (t~mL ~Ln)(TLTL)] -- 2n(3n + 1)(t~m R TR)(~[n TL) + 2n (7n + 2)[(~m
R
crmn TR)(t~n L TL) + (~mL °ran TL)(~n RTR)] •
(11)
Since the "new" 16 + 16 version is derived with a reducible compensating multiplet, this version is reducible. The seeming irreducibility comes from the "mixing-up" gauge condition (3) and the particular definition of fields (5). Finally we give explicitly the field-dependent gauge transformation which changes the "mixing-up" gauge conditions into the ordinary gauge conditions of the old minimal version [8]. Let us introduce C O and Z 0 fields,
C O = CA -2 ,
Z 0 =Z + 2CiT5 (A + i T 5 B ) - I x .
(12)
An advantage of using three variables is that C O is D, S, K-inert and Z 0 is S, K-inert. Then the desired gauge transformation is given by ,2 u = UKUSUD,
U D = exp [a D (/9)1 ,
with p = ~ (3n + 1) In C O ,
U S = exp [a s (~')] ,
with ~"= ~(3n + 1 ) c - l i T 5 Z 0 ,
U K = exp[6K(~m)] , with 7/m = e m (3 ~uP + ~ ~ ) .
(13)
The gauge fixing conditions (3) become
A' = UAU -1 = exp(p)A = 1,
B' = UBU -1 = exp(p)B = 0 ,
r
X =UxU - l = e x p ( ~ p ) [ X + ( A + i 7 5 B ) ~ " ] = 0 ,
t
b~=Ub~U -l=bu+aup+
1
~-2e~m~m
=0.
(14)
And other independent fields are redefined as follows : C' = e x p ( ~ )
Z ' = ~4- e x p [ 1~ ( 3 n + 5 ) 4 ] i 7 5 T,
,
B m = exp[2(3n + 1)4] [-W m - ] i exp(--4n4)(TLTm TR)] , P
A~ = (2/3n)G~
__
5r' = ~ exp[~(3n + 1)4] [S - ~-(3n + 1)TRTR] ,
[(3n + 1)/3n] e x p ( a n 4 ) Wu - ~ i(3n + 1) TLT~TR
4'u=exp[-~(3n+l)4][-~u+](3n+l)T~T],
e, m ' = e x p [ - ] ( 3 n + l ) 4 ] e g m
(15)
,2 Hexewe use the usual symbols 6D,S,K(.) although the parameters are not infinitesimal. 351
Volume 172, number 3,4
PHYSICS LETTERS B
22 May 1986
The remaining task is to rewrite the Poincar~ supersymmetry transformation 6~ ( t ) on the basis of the primed fields ¢ ' = UCU - 1 . The result is
~PQ(f,) = ~ld(Co(3n+l)14 ~,) + 6~lld (~-(3n
+ 1)C - 1
gi75omnZo).
(16)
Here 6~ ld is the usual Poincar6 supersymmetry transformation on the primed fields in the old minimal case. So eq. (16) explicitly shows that the complicated transformation laws 6~ ( t ) o f eq. (6) is in fact reducible to that o f the old minimal case if we rewrite all the fields by the primed ones. And the lagrangian (11) rewritten in terms o f the primed fields (15) shows that the real linear multiplet [C', Z ' , B m ] couples to the old minimal supergravity We would like to thank T. Kugo for valuable discussions.
Note added. After completing the manuscript we were informed by T. Kugo about a preprint [9] in which the reducibility o f 16 + 16 supergravity is pointed out. After submitting this paper, we received a preprint [10] from one o f the authors. One o f their gauge fixing conditions (8b) is quite different but the difference does not affect the reduced Poincar~ supergravity transformation laws and the lagrangian. References [1] K.S. Stelle and P.C. West, Phys. Lett. B 74 (1978) 330; S. Ferrara and P. van Nieuwenhuizen, Phys. Lett. B 74 (1978) 333. [2] M.F. Sohnius and P.C. West, Phys. Lett. B 105 (1981) 353. [3] P. Breitenlohner, Phys. Lett. B 67 (1977) 49; Nucl. Phys. B 124 (1977) 500. [4] M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Phys. Rev. D17 (1978) 3179; M. Kaku and P.K. Townsend, Phys. Lett. B 76 (1978) 54; P.K. Townsend and P. van Nieuwenhuizen, Phys. Rev. D19 (1979) 3166, 3592. [5] G. Girardi, R. Grimm, M. MiiUer and J. Wess, Phys. Lett. B 147 (1984) 81. [6] See for example: T. Kugo and S. Uehara, Nucl. Phys. B226 (1983) 49; T. Kugo and S. Uehara, Prog. Theor. Phys. 73 (1985) 235. [7] B. de Wit, R. Philippe and A. Van Proeyen, Nucl. Phys. B 219 (1983) 143. [8] T. Kugo and S. Uehara, Nucl. Phys. B226 (1983) 93. [9] W. Siegel, Maryland preprint #86-48. [10] C.S. Aulakh, J.-P. Derendinger and S. Ouvry, Paris preprint.
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